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PAPER Valley-momentum locking in a graphene superlattice with Y-shaped OPEN ACCESS Kekulé bond texture
RECEIVED 11 October 2017 O V Gamayun1,3, V P Ostroukh1, N V Gnezdilov1, İ Adagideli2 and C W J Beenakker1 REVISED 1 Instituut-Lorentz, Universiteit Leiden, PO Box 9506, 2300 RA Leiden, The Netherlands 12 January 2018 2 Faculty of Engineering and Natural Sciences, Sabanci University, Orhanli-Tuzla, 34956 Istanbul, Turkey ACCEPTED FOR PUBLICATION 3 Author to whom any correspondence should be addressed. 15 January 2018
PUBLISHED E-mail: [email protected] 7 February 2018 Keywords: graphene, Kekule distortion, valley-momentum locking, massless Dirac fermions
Original content from this work may be used under the terms of the Creative Abstract Commons Attribution 3.0 licence. Recent experiments by Gutiérrez et al (2016 Nat. Phys. 12 950) on a graphene–copper superlattice Any further distribution of have revealed an unusual Kekulé bond texture in the honeycomb lattice—a Y-shaped modulation of this work must maintain attribution to the weak and strong bonds with a wave vector connecting two Dirac points. We show that this so-called author(s) and the title of ‘Kek-Y’ texture produces two species of massless Dirac fermions, with valley isospin locked parallel or the work, journal citation and DOI. antiparallel to the direction of motion. In a magnetic field B, the valley degeneracy of the B-dependent Landau levels is removed by the valley-momentum locking but a B-independent and valley-degenerate zero-mode remains.
1. Introduction
The coupling of orbital and spin degrees of freedom is a promising new direction in nano-electronics, referred to as ‘spin-orbitronics’, that aims at non-magnetic control of information carried by charge-neutral spin currents [1–3]. Graphene offers a rich platform for this research [4, 5], because the conduction electrons have three distinct spin quantum numbers: in addition to the spin magnetic moment s=±1/2, there is the sublattice pseudospin σ=A, B and the valley isospin τ=K, K′. While the coupling of the electron spin s to its momentum p is a relativistic effect, and very weak in graphene, the coupling of σ to p is so strong that one has a pseudospin-momentum locking: the pseudospin points in the direction of motion, as a result of the helicity
operator p · s º+ppx ssx y y in the Dirac Hamiltonian of graphene. The purpose of this paper is to propose a way to obtain a similar handle on the valley isospin, by adding a term p · t to the Dirac Hamiltonian, which commutes with the pseudospin helicity and locks the valley to the direction of motion. We find that this valley-momentum locking should appear in a superlattice that has recently been realized experimentally by Gutiérrez et al [6, 7]: a superlattice of graphene grown epitaxially onto Cu(111), with the copper atoms in registry with the carbon atoms. One of six carbon atoms in each superlattice unit cell ( 33´ larger than the original graphene unit cell) have no copper atoms below them and acquire a shorter nearest-neighbor bond. The resulting Y-shaped periodic alternation of weak and strong bonds (see figure 1) is called a Kekulé-Y (Kek-Y) ordering, with reference to the Kekulé dimerization in a benzene ring (called Kek-O in this context) [7]. The Kek-O and KeK-Y superlattices have the same Brillouin zone, with the K and K′ valleys of graphene folded on top of each other. The Kek-O ordering couples the valleys by opening a gap in the Dirac cone [8–12], and it was assumed by Gutiérrez et althat the same applies to the Kek-Y ordering [6, 7]. While it is certainly possible that the graphene layer in the experiment is gapped by the epitaxial substrate (for example, by a sublattice-symmetry breaking ionic potential [13–15]),wefind that the Y-shaped Kekulé bond ordering by itself does not impose a mass on the Dirac fermions4. Instead, the valley degeneracy is broken by the helicity operator
4 That the Kek-Y bond ordering by itself preserves the massless nature of the Dirac fermions in graphene could already have been deduced from [15] (it is a limiting case of their equation (4)), although it was not noticed in the experiment [6]. We thank Dr Gutiérrez for pointing this out to us.
© 2018 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft New J. Phys. 20 (2018) 023016 O V Gamayun et al
Figure 1. Honeycomb lattices with a Kek-O or Kek-Y bond texture, all three sharing the same superlattice Brillouin zone (yellow hexagon, with reciprocal lattice vectors K). Black and white dots label A and B sublattices, black and red lines distinguish different bond strengths. The lattices are parametrized according to equation (4)(with f=0) and distinguished by the index ν=1+q−p modulo 3 as indicated. The K and K′ valleys (at the green Dirac points) are coupled by the wave vector GK=-+- Kof the Kekulé bond texture and folded onto the center of the superlattice Brillouin zone (blue point).
p · t, which preserves the gapless Dirac point while locking the valley degree of freedom to the momentum. In a magnetic field the valley-momentum locking splits all Landau levels except for the zeroth Landau level, which remains pinned to zero energy.
2. Tight-binding model
2.1. Real-space formulation A monolayer of carbon atoms has the tight-binding Hamiltonian
3 † Htab=-åå r,ℓ rrs+ ℓ +H.c.,() 1 r ℓ=1 describing the hopping with amplitude tr,ℓ between an atom at site ra=+nm1 a2 (n, m Î ) on the A sublattice (annihilation operator ar ) and each of its three nearest neighbors at rs+ ℓ on the B sublattice (annihilation operator b ). The lattice vectors are defined by s =-1 ( 3, 1), s =-1 ( 3, 1), s = ()0, 1 , rs+ ℓ 1 2 2 2 3 a131=-ss, a232=-ss. All lengths are measured in units of the unperturbed C–C bond length a0≡1. For the uniform lattice, with tr,ℓ º t0, the band structure is given by [16] 3 iks· Et()kkk= ∣()∣ee,e. () = 0 å ℓ () 2 ℓ=1 There is a conical singularity at the Dirac points K =2 p 31,3(), where E ()K = 0. For later use we 9 note the identities
2i3p ee()kkK=+ (3e+- ) = e ( kKK ++ ) . () 3
2 New J. Phys. 20 (2018) 023016 O V Gamayun et al
The bond-density wave that describes the Kek-O and Kek-Y textures has the form
ii()··pqKKsGr++ ttr,0ℓ =+D12Ree[]+-ℓ ⎡ ⎤ =+D12cos⎣fp +2 ()mnN - + ⎦ , () 4 a 0 3 ℓ
NqNpNpqpq123=-,, =- = + ,,. Î 3 () 4 b
The Kekulé wave vector
GKº-= K 4 p 31,0() () 5 +-9
if couples the Dirac points. The coupling amplitude D =D0e may be complex, but the hopping amplitudes tr,ℓ are real in order to preserve time-reversal symmetry. (We note that our definition of Δ differs by a factor 3 from that of [8].) As illustrated in figure 1, the index
n =+-1mod3qp () 6 distinguishes the Kek-O texture (ν=0) from the Kek-Y texture (ν= ±1). Each Kekulé superlattice has a 2π/3 rotational symmetry, reduced from the 2π/6 symmetry of the graphene lattice. The two ν=±1 Kek-Y textures are each others mirror image5.
2.2. Transformation to momentum space The Kek-O and Kek-Y superlattices have the same hexagonal Brillouin zone, with reciprocal lattice vectors K—smaller by a factor 1 3 and rotated over 30° with respect to the original Brillouin zone of graphene (see figure 1). The Dirac points of unperturbed graphene are folded from the corner to the center of the Brillouin zone and coupled by the bond-density wave. To study the coupling we Fourier transform the tight-binding Hamiltonian (1),
†† Habpqab()kk=-ee ()k k -D ( kKK ++- + )kG+ k † -D*e()kK -pqab+- - KkG- k +H.c. () 7
The momentum k still varies over the original Brillouin zone. In order to restrict it to the superlattice Brillouin zone we collect the annihilation operators at k and k G in the column vector caaabbbk= () k,,,,, kG-+ kG k kG -+ kGand write the Hamiltonian in a 6×6 matrix form: ⎛ 0 ()k ⎞ †⎜ n ⎟ Hc()k =-k † ck,8() a ⎝ n()k 0 ⎠ ⎛ ˜˜* ⎞ ⎜ ee011DDnn+-- e⎟ n = ⎜DD˜˜*ee e⎟,8()b ⎜ 11--nn⎟ ˜˜ ⎝ DDeeenn--11* ⎠ ˜ 2ip ()pq+ 3 D=e, Deen =()kG +nc , () 8 where we used equation (3).
3. Low-energy Hamiltonian
3.1. Gapless spectrum The low-energy spectrum is governed by the four modes uaabbk= () kGkGkGkG-+-+,,, , which for small k lie near the Dirac points at G. (We identify the K valley with +G and the K′ valley with -G.) Projection onto this subspace reduces the six-band Hamiltonian (8) to an effective four-band Hamiltonian, ⎛ ⎞ ⎛ ⎞ 0 hn eeD˜ Hu=-†⎜ ⎟uh,. =⎜ -1 n⎟ () 9 eff k ⎝ † ⎠ k n ˜ hn 0 ⎝D*ee-n 1 ⎠ 2 Corrections to the low-energy spectrum from virtual transitions to the higher bands are of order D0. We will include these corrections later, but for now assume Δ0=1 and neglect them.
5 There are three sets of integers p, q Î 3 for a given index n =+-1mod3qp , corresponding to textures on the honeycomb lattice that are translated by one hexagon, or equivalently related by a ±2π/3 phase shift of Δ.
3 New J. Phys. 20 (2018) 023016 O V Gamayun et al
Figure 2. Dispersion relation near the center of the superlattice Brillouin zone, for the Kek-O texture (blue dashed curves) and for the ˜ Kek-Y texture (black solid). The curves are calculated from the full Hamiltonian (8) for ∣D=D=∣ 0 0.1.
The k-dependence of en may be linearized near k = 0, 2 ee0010==++3,tvkkk ()() xy i order , () 10 with Fermi velocity vta= 3 . The corresponding 4-component Dirac equation has the form 0 2 00 ⎛ ⎞ ⎛ ⎞ ⎛ ˜ ⎞ YK¢¢YK ⎜vQ0p · s D n ⎟ ⎜ ⎟ = E⎜ ⎟,,= ⎜ ⎟ () 11a ⎝ Y ⎠ ⎝ Y ⎠ ˜ * † K K ⎝ D Qvn 0p · s⎠ ⎛ ⎞ ⎛ ⎞ -yBK, ¢ yAK, Y=K¢ ⎜ ⎟,,Y=K ⎜ ⎟ () 11b ⎝ yAK, ¢ ⎠ ⎝yBK, ⎠ ⎛ ⎞ ⎧3if0,t sn= e*-n 0 0 z Qn = ⎜ ⎟ = ⎨ ()11c ⎝ 0 -en⎠ ⎩vp00()∣∣nsnxy-=iif1. p
The spinor ΨK contains the wave amplitudes on the A and B sublattices in valley K and similarly YK¢ for valley K′, but note the different ordering of the components [17]6. We have defined the momentum operator σ pr=-i ¶ ¶ , with p · s =+ppx ssx y y. The Pauli matrices sxyz,,ss, with 0 the unit matrix, act on the sublattice degree of freedom. For the Kek-O texture we recover the gapped spectrum of Kekulé dimerized graphene [8],
2 2 2 2 Ev=+D=0∣∣p (3 t00 ) forn 0. ( 12 )
The Kek-Y texture, instead, has a gapless spectrum,
2 2 22 Ev =D0 ()∣∣∣∣1,for1,0 p n = () 13 consisting of a pair of linearly dispersing modes with different velocities v00()1 D . The two qualitatively different dispersions are contrasted in figure 2.
3.2. Valley-momentum locking The two gapless modes in the Kek-Y superlattice are helical, with both the sublattice pseudospin and the valley ˜ isospin locked to the direction of motion. To see this, we consider the ν=1 Kek-Y texture with a real D =D0. (Complex D˜ and ν=−1 are equivalent upon a unitary transformation.) The Dirac Hamiltonian (11) can be written in the compact form
=Ä+Ävvst(·)ppstts00 (·),14 () 7 with the help of a second set of Pauli matrices τx, τy, τz and unit matrix τ0 acting on the valley degree of freedom . The two velocities are defined by vσ=v0 and vτ=v0Δ0.
6 The ordering of the spinor components in equation (11b) is the so-called valley-isotropic representation of Dirac fermions. 7 For reference, we note that the unitary transformation from Y =-( yyyyBK,,,,¢¢,,, AK AK BK) to Y¢=(yyyyBK,,,,¢¢,,, AK AK BK) transforms Hv=Ä+Äst(·)ppstts00 v (·) into =-vvst(·)ppst Ätszz + Ä (·).
4 New J. Phys. 20 (2018) 023016 O V Gamayun et al
Figure 3. Landau levels in the Kek-Y superlattice (Δ0=0.1, f=0, ν=1). The data points are calculated numerically [22] from the tight-binding Hamiltonian (1) with bond modulation (4). The lines are the analytical result from equations (18) and (19) for the first few Landau levels. Lines of the same color identify the valley-split Landau level, the zeroth Landau level (red line) is not split.
An eigenstate of the current operator
jpvvaa=¶ ¶ =sast Ä00 + t st Ä a ()15 with eigenvalue vvst is an eigenstate of σα with eigenvalue +1 and an eigenstate of τα with eigenvalue ±1. (The two Pauli matrices act on different degrees of freedom, so they commute and can be diagonalized independently.) This valley-momentum locking does not violate time-reversal symmetry, since the time- reversal operation in the superlattice inverts all three vectors p, s, and t, and hence leaves unaffected8:
()()styyÄÄ=* st yy .16 () The valley-momentum locking does break the sublattice symmetry, since no longer anticommutes with
σz, but another chiral symmetry involving both sublattice and valley degrees of freedom remains:
()stzzÄ=-Ä () st zz.17 ()
3.3. Landau level quantization A perpendicular magnetic field B in the z-direction (vector potential A in the x–y plane), breaks the time-reversal symmetry (16) via the substitution prAr -¶¶+i e () ºP. The chiral symmetry (17) is preserved, so the Landau levels are still symmetrically arranged around E=0, as in unperturbed graphene. Because the two helicity operators P · s and P · t do not commute for A ¹ 0, they can no longer be diagonalized independently. In particular, this means the Landau level spectrum is not simply a superposition of two spectra of Dirac fermions with different velocities. It is still possible to calculate the spectrum analytically (see appendix A).Wefind Landau levels at energies +- + - Enn,,EEE-- n , n, n =¼0, 1, 2, , given by 2412- EEnn =+++B [()()21 1 nnvvv 14st ¯ ]() , 18 fi 22 with the de nitions vvv¯ =+stand EB = veB¯ . 9 + - = In unperturbed graphene all Landau levels have a twofold valley degeneracy : Enn= E +1 for vτ 0. This -- includes the zeroth Landau level: E00==-0 E . A nonzero vτ breaks the valley degeneracy of all Landau levels - fi at E ¹ 0, but a valley-degenerate zero-mode E0 = 0 remains, see gure 3. The absence of a splitting of the zeroth Landau level can be understood as a topological protection in the context of an index theorem [18–21], which requires that either P+ ºPxy +i P or P- ºPxy -i P has a zero- mode. If we decompose =P+-SS +P -+, with Sv =+st()()ssxyii v tt xy, we see that both S+ and
8 The time-reversal operation =Ä()styy from equation (16)(with complex conjugation) squares to +1 because the electron spin is not explicitly included. If we do include it, we would have =ÄÄ()syyyst, which squares to −1 as expected for a fermionic quasiparticle. The combination of the time-reversal symmetry (16) and the chiral symmetry (17) places the superlattice in the BDI symmetry classification of topological states of matter. 9 The Landau levels also have a twofold spin degeneracy, which could be resolved by the Zeeman energy but is not considered here.
5 New J. Phys. 20 (2018) 023016 O V Gamayun et al
10 ()1 ()2 S− have a rank-two null space , spanned by the spinors y and y .SoifP f = 0, a twofold degenerate ()1 ()2 zero-mode of is formed by the states fy and fy . All of this is distinctive for the Kek-Y bond order: for the Kek-O texture it is the other way around—the Landau levels have a twofold valley degeneracy except for the nondegenerate Landau level at the edge of the band gap11.
4. Effect of virtual transitions to higher bands
So far we have assumed D0 1, and one might ask how robust our findings are to finite-Δ0 corrections, involving virtual transitions from the e1 bands near E=0 to the e0 band near E=3t0. We have been able to include these to all orders in Δ0 (see appendix B), and find that the entire effect is a renormalization of the velocities vσ and vτ in the Hamiltonian (14), which retains its form as a sum of two helicity operators. For real
Δ=Δ0 the renormalization is given by vσ=v0ρ+, vτ=v0ρ− with ⎛ ⎞ 1 ⎜ 12+D0 ⎟ r =-D()1 0 1. () 19 2 ⎜ 2 ⎟ ⎝ 12+D0 ⎠
if For complex D =D0e the nonlinear renormalization introduces a dependence on the phase f modulo 2π/3. What this renormalization shows is that, as expected for a topological protection, the robustness of the zeroth Landau level to the Kek-Y texture is not limited to perturbation theory—also strong modulations of the bond strength cannot split it away from E=0.
5. Pseudospin-valley coupling
In zero magnetic field the low-energy Hamiltonian (14) does not couple the pseudospin σ and valley τ degrees of freedom. A s Ä t coupling is introduced in the Kek-Y superlattice by an ionic potential μY on the carbon atoms that line up with the carbon vacancies—the atoms located at each center of a red Y in figure 1. We consider this ˜ effect for the ν=1 Kek-Y texture with a real D =D0. The ionic potential acts on one-third of the A sublattice sites, labeled rY. (For ν=−1 it would act on one- third of the B sublattice sites.) Fourier transformation of the on-site contribution m aa† to the tight- Y år Y rY r Y binding Hamiltonian (1) gives with the help of the lattice sum å e20ikr· Y µ+-++dd()()()kkGkG d () rY the momentum-space Hamiltonian
⎛ M ()k ⎞ †⎜ Y1⎟ Hc()k =-k † cak,21() ⎝1()k 0 ⎠
⎛111⎞ ⎜ ⎟ MbY =-mY ⎜111⎟.21() ⎝111⎠
The 1 block is still given by equation (8). The additional MY-block breaks the chiral symmetry. Projection onto the subspace spanned by uaabbk= () kGkGkGkG-+-+,,, gives the effective Hamiltonian ⎛ ⎞ mhY1 Hu=-†⎜ ⎟um, =-m 11.22() eff k ⎝ † ⎠ k Y Y ()11 h1 0 The corresponding Dirac Hamiltonian has the form (11) with an additional s Ä t coupling, =Ä+Ä+vv(·)ppstts (·) 1 m st00 2 Y +Ä+Ä-Ä1 ms() t s t s t.23 () 2 Y xxyyzz
10 fi ()1 If we de ne the eigenstates ∣ab, ñ by sz∣∣ab,,ñ= aabñ, tabz∣∣,,ñ= bab ñ, then S+ annihilates y+ = ∣1, 1ñand ()2 ()1 ()2 y+ =-vvts∣∣1, 1 ñ- 1, - 1ñ, while S− annihilates y- =-∣ 1, -ñ 1 and y- =-ñ--ñvvts∣∣1, 1 1, 1 . 11 2 2 2 = K In a Kek-O superlattice the Landau levels are given by En =D+()32tneBv00 0 , n 0, 1, 2, , with a twofold valley degeneracy for n 1 and a nondegenerate zeroth Landau level at D3t00.
6 New J. Phys. 20 (2018) 023016 O V Gamayun et al
fi Figure 4. Effect of an on-site potential mY on the Kek-Y band structure of gure 2. The three bands that intersect linearly and quadratically at the center of the superlattice Brillouin zone form the ‘spin-one Dirac cone’ of [14, 15]. The curves are calculated from ( ) the full Hamiltonian 21 for D=0 0.1 =mY.
Figure 5. Orientation of the expectation value of the pseudospin sss= ( xy, ) (left panel) and valley isospin ttt= ( xy, ) (right panel) in the two bands (24) at E>0. The pseudospin points in the direction of motion in both bands, while the valley isospin is locked parallel to the direction of motion in one band (red arrows) and antiparallel in the other band (blue arrows).
The energy spectrum,
()1 Evv =()∣∣st - p , ()2 22 2 Evv =mmY ()∣∣st +p +Y ,24 () has two bands that cross linearly in p at E=0, while the other two bands have a quadratic p-dependence (see figure 4). The pseudospin and valley isospin orientation for the two bands is illustrated in figure 5. ()1 ()1 ()2 = The three bands E+ , E- , E- that intersect at p 0 are reminiscent of a spin-one Dirac one. Such a dispersion is a known feature of a potential modulation that involves only one-third of the atoms on one sublattice [14, 15]. The spectrum remains gapless even though the chiral symmetry is broken. This is in contrast to the usual staggered potential between A and B sublattices, which opens a gap via a szzÄ t term [16].
6. Discussion
In summary, we have shown that the Y-shaped Kekulé bond texture (Kek-Y superlattice) in graphene preserves the massless character of the Dirac fermions. This is fundamentally different from the gapped band structure resulting from the original Kekulé dimerization [8–11] (Kek-O superlattice), and contrary to expectations from its experimental realization [6, 7]. The gapless low-energy Hamiltonian =+vvstpp··stis the sum of two helicity operators, with the momentum p coupled independently to both the sublattice pseudospin s and the valley isospin t. This valley- momentum locking is distinct from the coupling of the valley to a pseudo-magnetic field that has been explored as an enabler for valleytronics [23], and offers a way for a momentum-controlled valley precession. The broken valley degeneracy would also remove a major obstacle for spin qubits in graphene [24]. A key experimental test of our theoretical predictions would be a confirmation that the Kek-Y superlattice has a gapless spectrum, in stark contrast to the gapped Kek-O spectrum. In the experiment by Gutiérrez et alon a graphene/Cu heterostructure the Kek-Y superlattice is formed by copper vacancies that are in registry with one out of six carbon atoms [6, 7]. These introduce the Y-shaped hopping modulations shown in figure 1, but in addition will modify the ionic potential felt by the carbon atom at the center of the Y. Unlike the usual staggered
7 New J. Phys. 20 (2018) 023016 O V Gamayun et al potential between A and B sublattices, this potential modulation in an enlarged unit cell does not open a gap [14, 15]. We have also checked that the Dirac cone remains gapless if we include hoppings beyond nearest neighbor. All of this gives confidence that the gapless spectrum will survive in a realistic situation. Further research in other directions could involve the Landau level spectrum, to search for the unique feature of a broken valley degeneracy coexisting with a valley-degenerate zero-mode. The graphene analogs in optics and acoustics [25] could also provide an interesting platform for a Kek-Y superlattice with a much stronger amplitude modulation than can be realized with electrons.
Acknowledgments
We have benefited from discussions with A Akhmerov, V Cheianov, J Hutasoit, P Silvestrov, and D Varjas. This research was supported by the Netherlands Organization for Scientific Research (NWO/OCW) and an ERC Synergy Grant.
Appendix A. Calculation of the Landau level spectrum in a Kek-Y superlattice
We calculate the spectrum in a perpendicular magnetic field of a graphene sheet with a Kekulé-Y bond texture. We start by rewriting the Hamiltonian (14), with P =+pAe , in the form =P1 SS +P1 +ms Ä t ,A1() 2 +-2 -+ zz in terms of the raising and lowering operators
P=PPxyi,ssstt = xy i, = xy i, t
Sv=Ä+Ästst00 v st .A2()
The chiral-symmetry breaking term μσz⊗τz that we have added will serve a purpose later on. We know that the Hermitian operator Ω=Π+Π− has eigenvalues wn = 2neB , n=0, 1, 2, K, in view of the commutator [PP=-+,2] eB. So the strategy is to express the secular equation det()E -= 0 in a form 2 2 that involves only the mixed products P+-P , and no P+ or P-. This is achieved by means of a unitary transformation, as follows. We define the unitary matrix ⎡ ⎤ U =+Äexp⎣ 1 ips() s t⎦ () A3 4 0 zy and reduce the determinant of a 4×4 matrix to that of a 2×2 matrix:
det()-=EU det † () - EU ⎛ ⎞ -+ERm † = det ⎜ ⎟ ⎝ RE--m⎠ ⎧ det()ERRE22--mm† if ¹ , = ⎨ ()A4 ⎩det()ERRE22--mm† if ¹- ,
⎛-Pvv P⎞ withR = ⎜ ts--⎟ .() A5 ⎝-Pvvst++ P⎠
† † The matrix product RR is not of the desired form, but R R is,
⎛ 22 ⎞ vvPP-+ + PP +- - vvst() PP -+ +PP +- RR† = ⎜ st ⎟,A6() 22 ⎝-vvst() PP-+ +PP +- vst PP +- + v PP -+⎠ involving only P+-P=W and P-+P=W+w1. Hence the determinant is readily evaluated for E ¹-m, ¥ ⎛ 222 2 ⎞ Evvvv--mw¯ - wst()2 ww + det()( -=EE det22 --m RR† ) = det ⎜ nns 11⎟,A7() 222 2 n=0 ⎝ vvst()2wwnn+---1 E m v¯ w vt w1⎠ 22 where we have abbreviated vvv¯ =+st. Equating the determinant to zero and solving for E we find four sets of energy eigenvalues +- + - Enn,,EEE-- n , n, given by
8 New J. Phys. 20 (2018) 023016 O V Gamayun et al
()EvvvvEnnnvvv 22-=mw +1 w¯¯2 1 w2 42 +()421114. ww =2 [ ++ ( + )()24¯- ] n ()nnn2 1 2 1 st+1 B st ()A8
In the second equation we introduced the energy scale EB = vl¯ m, with lm = eB the magnetic length. The - B-independent level E0 = m becomes a zero-mode in the limit m 0. As a check on the calculation, we note that for μ=0, vτ=0 we recover the valley-degenerate Landau level spectrum of graphene [16], -+- EvlnEEn ==() s m 2,nn+1 . () A9
Another special case of interest is μ=0, vvvst=º0, when the two modes of Dirac fermions have velocities vvst equal to 0 and 2v0. From equation (A8) we find the Landau level spectrum -+ EEvlnnn==0, 2() 0 m 2 + 1. () A10
The mode with zero velocity remains B-independent, while the mode with velocity 2v0 produces a sequence of Landau levels with a 1/2 offset in the n-dependence.
Appendix B. Calculation of the low-energy Hamiltonian to all orders in the Kek-Y bond modulation
We seek to reduce the six-band Hamiltonian (8) to an effective 4×4 Hamiltonian that describes the low-energy spectrum near k = 0. For D0 1we can simply project onto the 2×2 lower-right subblock of n, which for the ∣n∣ = 1 Kek-Y bond modulation vanishes linearly in k. This subblock is coupled to the e0 band near E = 3t0 by matrix elements of order Δ0, so virtual transitions to this higher band contribute to the low-energy spectrum 2 Δ in order D0. We will now show how to include these effects to all order in 0. One complication when we go beyond the small-Δ0 regime is that the phase f of the modulation amplitude can no longer be removed by a unitary transformation. As we will see, the low-energy Hamiltonian depends on f modulo 2π/3—so we do not need to distinguish between the phase of D˜ =De2ip ()pq+ 3 and the phase of Δ. The choice between n =1still does not matter, the two Kek-Y modulations being related by a mirror symmetry. For definiteness we take ν=+1. We define the unitary matrix ⎛10 0⎞ F 0 ⎛ 0⎞ ⎜ ⎟ Va= ⎜⎟, F= 0e-if 0 ,B1() ()0 F ⎝ 0 ⎠ ⎜ ⎟ ⎝00eif⎠ ⎛ 22-D -D 2⎞ 1 ⎜ 00⎟ = ⎜21D+000DD 1 -⎟,B1()b 2D0 ⎝21D-000DD 1 +⎠
2 with D0 =+D120 and evaluate ⎛ ⎞ ⎛ ⎞ 0 1 0 ˜1 VV†⎜ ⎟ = ⎜ ⎟,B2() a ⎝ ††⎠ ˜ 1 0 ⎝1 0 ⎠ ⎛ ⎞ erere* ⎜D00 0 -1 0 1⎟ ˜ † ⎜ * ⎟ 11==0 re+ -11 re- ,B2()b ⎜ ⎟ * ⎝ 0 re- -11 re+ ⎠
1 2 -3if r =-D+D[()]()120 DD0 e00 1 , B2 c 2D0
D0 3if r0 = ()2e+D0 . () B2d D0 ˜ The matrix elements that couple the lower-right 2×2 subblock of 1 to e0 are now of order k, so the effect on 2 the low-energy spectrum is of order k and can be neglected—to all orders in Δ0. The resulting effective low-energy Hamiltonian has the 4×4 form (9), with h1 replaced by ⎛ ⎞ re re* ⎜ + -11- ⎟ h1 = ⎜ ⎟.B3() * ⎝re- -11 re+ ⎠
iq × The phases of r= ∣∣r e can be eliminated by one more unitary transformation, with the 4 4 diagonal matrix
9 New J. Phys. 20 (2018) 023016 O V Gamayun et al
Figure B1. Velocities vvv1 =+stand vvv2 =-stof the two gapless modes in the Kek-Y superlattice, as a function of the bond modulation amplitude Δ0 for two values of the modulation phase f. The f-dependence modulo 2π/3 appears to second order in Δ0. The curves are calculated from equation (B7). Note that positive and negative values of v1, v2 are equivalent.
if Figure B2. Kek-Y superlattice with a complex bond amplitude D=e D0, according to equation (4) with ν=1. The three colors of the bonds refer to three different bond strengths, adding up to 3t0. For f=0 two of the bond strengths are equal to t00(1 -D) and fi f=π/ the third equals t00(12+D). This is the case shown in gure 1. For 6 the bond strengths are equidistant: t00(13-D ), t0, Δ fi and t00(13+D ). The value of 0 where a bond strength vanishes shows up in gure B1 as a point of vanishing velocity.
Q=diage,e,e()iiiiqqqq-++-+ ,1, () B4 which results in
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 h1 0 h˜ ∣∣re ∣∣ re †⎜ ⎟ ⎜ 1⎟ ˜ ⎜ + -11- ⎟ QQ==⎜ † ⎟ ⎜ † ⎟,.h1 () B5 ⎝ ˜ ⎠ ˜ ⎝∣∣re- ∣∣ re⎠ h1 0 ⎝h1 0 ⎠ - 11+
Finally, we arrive at the effective Hamiltonian (14), with renormalized velocities:
=Ä+Ä=vvvvvvstst(·)ppstts00 (·),,, ∣∣ r+- 0 = ∣∣ r 0 () B6
1 rf2 = +D4 -D2 +D3 - ∣∣ 2 (130 DD0 ( 130 ) 20 (0 2cos3. ) ) () B7 2D0
To third order in Δ0 we have
vv=1 -3 D-2 1 D3 cos 3fff , vv =D-D3 2 cos 3 +1 D3()()() 1 - 9 cos 6 + D4 . B8 st0 2 0 2 0 002 0 16 0 0
10 New J. Phys. 20 (2018) 023016 O V Gamayun et al
For real Δ, when f=0 and ρ± is real, equation (B7) simplifies to ⎛ ⎞ 1 ⎜ 12+D0 ⎟ r =-D()1 0 1. () B9 2 ⎜ 2 ⎟ ⎝ 12+D0 ⎠ The velocities of the two Dirac modes are then given by
()()112-D00 + D vv10=+=st v v , 2 12+D0
vvvv200=-=st ()1. -D () B10 if More generally, for complex D =D0e both v1 and v2 become f-dependent to second order in Δ0, see figure B1. Note that the asymmetry in D0 vanishes for f=π/6. For this phase the superlattice has three different bond strengths (see figure B2) that are symmetrically arranged around the unperturbed value t0.
ORCID iDs
C W J Beenakker https://orcid.org/0000-0003-4748-4412
References
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